What is standard deviation?
Standard deviation tells you how far the numbers typically sit from their mean. It is the square root of the variance, so it is in the same units as your numbers: days, inches or points. For data shaped like a bell curve, about two-thirds of the numbers fall within one standard deviation of the mean.
Source CDC · Principles of Epidemiology, Lesson 2, Section 7: Measures of spread ↗NIST/SEMATECH e-Handbook of Statistical Methods · 1.3.5.6 Measures of scale ↗
Sample or population standard deviation?
Use the sample standard deviation unless you have every value there is. A sample, like 30 patients from one hospital, divides by one less than how many numbers there are (n − 1), which makes the answer a little bigger. The population version divides by n, and fits only when the list is complete, like all 10 tools a factory made. This calculator always shows both, so you can see the difference.
| Your numbers are | Excel formula |
|---|---|
| A sample | =STDEV.S(A2:A11) |
| The whole population | =STDEV.P(A2:A11) |
Source Microsoft Support · STDEV.S function (sample), with example ↗Microsoft Support · STDEV.P function (whole population), with example ↗
How to calculate standard deviation
CDC gives five steps. Here they are with your numbers.
Here it is with your numbers:
- Find the mean: the 5 numbers add up to 125, and 125 ÷ 5 = 25.
- Subtract the mean from each number and square the result: (27 − 25)² gives 4, (31 − 25)² gives 36, (15 − 25)² gives 100, (30 − 25)² gives 25, (22 − 25)² gives 9.
- Add the squares: 4 + 36 + 100 + 25 + 9 = 174.
- Divide by 4, one less than how many numbers there are: 174 ÷ 4 = 43.5. This is the variance.
- Take the square root of the variance: √43.5 = 6.6.
It means a typical number is about 6.6 away from the mean of 25, some closer and some further.
Each number’s distance from the mean
| Number | Minus the mean | Squared |
|---|---|---|
| 27 | 2 | 4 |
| 31 | 6 | 36 |
| 15 | −10 | 100 |
| 30 | 5 | 25 |
| 22 | −3 | 9 |
| Total 125 | 0 | 174 |
Standard deviation examples
| Numbers | Standard deviation |
|---|---|
| 27, 31, 15, 30, 22 (days) | Sample 6.6 |
| 1345, 1301 … 1299 (10 tools) | Sample 27.46 |
| The same 10 tools, as the whole population | Population 26.05 |
| 2, 0, 3, 1, 0, 1, 2, 2, 4, 8, 1, 3, 3, 12, 1, 6, 2, 5, 1 (doses) | Sample 3 |
FAQs
How do you find the standard deviation?
Find the mean, subtract it from each number and square the results, add the squares, divide by one less than how many numbers there are, then take the square root. For 27, 31, 15, 30 and 22 days, the mean is 25, the squares add up to 174, 174 ÷ 4 is 43.5, and the square root of 43.5 is 6.6 days.
Source CDC · Principles of Epidemiology, Lesson 2, Section 7: Measures of spread ↗
What is standard deviation?
Standard deviation is a measure of how spread out numbers are around their mean. It is the square root of the variance, which puts the spread back in the same units as the numbers themselves.
Source NIST/SEMATECH e-Handbook of Statistical Methods · 1.3.5.6 Measures of scale ↗
What does standard deviation mean?
It shows how widely or tightly the numbers are spread around the average. For data shaped like a bell curve, about two-thirds (68.3%) fall within one standard deviation of the mean, and 95.5% within two.
Source CDC · Principles of Epidemiology, Lesson 2, Section 7: Measures of spread ↗
How do you calculate standard deviation in Excel?
Use =STDEV.S(A2:A11) when your numbers are a sample, and =STDEV.P(A2:A11) when they are the whole population. On Microsoft’s breaking-strength example, STDEV.S gives 27.46391572 and STDEV.P gives 26.05455814.
Source Microsoft Support · STDEV.S function (sample), with example ↗
Sources
- CDC: Principles of Epidemiology, Lesson 2, Section 7: Measures of spread
- NIST/SEMATECH e-Handbook of Statistical Methods: 1.3.5.6 Measures of scale
- Microsoft Support: STDEV.S function (sample), with example
- Microsoft Support: STDEV.P function (whole population), with example